Page:Aether and Matter, 1900.djvu/213

and

$\epsilon ^{\frac {1}{2}}\left(\epsilon ^{-{\frac {1}{2}}}a,\ b+4\pi vh,\ c-4\pi vg\right)$

had at a time const. $+vx/c^{2}$ before the instant considered when the scale of time is enlarged in the ratio $\epsilon ^{\frac {1}{2}}$.

As both the electric and magnetic vectors of radiation lie in the wave-front, it follows that in the two correlated systems, fixed and moving, the relative wave-fronts of radiation correspond, as also do the rays which are the paths of the radiant energy relative to the systems. The change of the time variable, in the comparison of radiations in the fixed and moving systems, involves the Doppler effect on the wave-length.

The Correlation between a stationary and a moving Medium, as regards trains of Radiation

113. Consider the aethereal displacement given by

$(f_{1},\ g_{1},\ h_{1})=(L,\ M,\ N)\ F\ (lx_{1}+my_{1}+nz_{1}-pt),$

which belongs to a plane wave-train advancing, along the direction (l, m, n) with velocity V, or c/μ where μ is refractive index, equal to

$p(l^{2}+m^{2}+n^{2})^{-{\frac {1}{2}}},$

in the material medium at rest referred to coordinates $(x_{1},\ y_{1},\ z_{1})$. In the corresponding wave-train relative to the same medium in motion specified by coordinates (x, y, z), and considered as shrunk in the above manner as a result of the motion, the vectors (f, g, h) and (a, b, c) satisfy the relation

$\epsilon ^{\frac {1}{2}}\left(\epsilon ^{-{\frac {1}{2}}}f,\ g-{\frac {v}{4\pi c^{2}}}c,\ h+{\frac {v}{4\pi c^{2}}}b\right)$

$=(L,\ M,\ N)\ F\left\{l\epsilon ^{\frac {1}{2}}x+my+nz-p\epsilon ^{-{\frac {1}{2}}}\left(t-{\frac {v}{c^{2}}}\epsilon x\right)\right\}$

$=(L,\ M,\ N)\ F\left\{\left(l\epsilon ^{\frac {1}{2}}+{\frac {pv}{c^{2}}}\epsilon ^{\frac {1}{2}}\right)x+my+nz-p\epsilon ^{-{\frac {1}{2}}}t\right\}.$}}

As the wave-train in the medium at rest is one of transverse displacement, so that the vectors $(f_{1},\ g_{1},\ h_{1})$ and $(a_{1},\ b_{1},\ c_{1})$ are both in the wave-front, the same is therefore true for the vectors (f, g, h) and (a, b, c) in the correlative wave-train in the moving system, as was in fact to be anticipated from the